1. Field of the Invention
This invention relates to a method for shortening response time of a logarithmic measuring apparatus to be used in the field of instrumentation of the nuclear reactors, critical assemblies and particle accelerators. More particularly, a method to reduce response time of a logarithmic measuring apparatus in which when the input signal into a logarithmic amplifier is current signal and when an electrostatic capacitance of some measure to ground, that cannot be disregarded, exists in a signal source and/or a signal line that connects the signal source to the logarithmic amplifier, a resistor of a specific resistance value is inserted between the signal source or the signal line and input terminals of the logarithmic amplifier.
2. Description of the Prior Art
An apparatus of the prior art to measure the logarithmic power of the nuclear reactors, critical assemblies and particle accelerators is generally shown in FIG. 1, in its construction.
In FIG. 1, 1 denotes nuclear radiation to be measured, 2 an ionization chamber for the nuclear radiation, 3 a signal cable including sheath and core conductor for transmitting output current of the ionization chamber, 4 a high voltage cable, 5 a high voltage power supply, 6 a logarithmic amplifier of current input type, which is usually referred to as Log-N amplifier, 7 a logarithmic output terminal and 8 a signal line of the core conductor of the signal cable. The output signal from the logarithmic output terminal is connected to a recorder or a meter to indicate the logarithmic output and also the output signal is connected to a period amplifier to be used for measuring the reactor period.
FIG. 2 shows an electrically equivalent circuit of the construction of the circuit shown in FIG. 1. A constant-current supply 11 is a power source that generates ionization chamber current I. An electrostatic capacitor 12 between electrodes of the ionization chamber, whose capacitance being represented by C.sub.d, and an electrostatic capacitor 13 of the signal line to ground, whose capacitance being represented by C.sub.c, are connected in parallel to the input circuit of logarithmic amplifier 6 and the sum of C.sub.d and C.sub.c is hereinafter referred to as an input capacitance C.sub.i. Coaxial cables of about 30 m to 50 m length are frequently used as signal cable 8 so that the value of input capacitance C.sub.i grows sometimes around 3000 pF. The basic construction of logarithmic amplifier 6 of current input type includes a logarithmic diode 14, a negative feedback capacitor 15 that is used to maintain the operational stability of the circuit and an operational amplifier 16 of high input resistance type. Some silicon diodes are recently used as logarithmic diode 14 widely. An amplification gain A for the D.C. signal, hereinafter referred to as D.C. gain, of operational amplifier 16 is made sufficiently large as compared with unity, i.e. 1, and the phase compensation is made so that the gain reduces at a rate of -20 db/decade in high frequency region over a high-cut-off frequency f.sub.c and phase lag does not exceed 90.degree.. Accordingly, a unity gain frequency f.sub.T of the operational amplifier is represented as follows: EQU f.sub.T =Af.sub.c.
The transfer function Go (S) of operational amplifier 16 is, therefore, represented by the following equation (1) as well known as the transfer function of the ordinary phase-compensated-operational-amplifier. EQU Go(S)=(-A/1+.tau.s) (1)
where EQU .tau.=1/2.pi.f.sub.c =A/2.pi.f.sub.T ( 2)
S is a complex variable in Laplace transformation.
It is an indispensable condition to insert negative feedback capacitor 15 having more than a definite value in the logarithmic amplifier circuit of the prior art in order that no instability of response is appeared such as oscillation, ringing and overshoot etc. in the circuit. Therefore, since the capacitance C.sub.f of negative feedback capacitor 15 is not made so small because of stability condition that is mentioned above, the time constant of response becomes large and it has been impossible to obtain rapid response characteristics in a region of small input current. The relationship between a minimum feedback capacitance C.sub.fmin that is necessary for stabilizing the circuit of the prior art and the response time of the circuit is quantitatively shown in the following explanation.
When the ionization chamber current I becomes I+i(t) after varying by i(t) under a condition of .vertline.i(t).vertline.&lt;&lt;.vertline.I.vertline., if it is assumed that i. the voltage variation at a connection point of signal line 8, logarithmic diode 14, feedback capacitor 15 and the input of operational amplifier 16 all shown in FIG. 2 is e.sub.1 (t), ii. the voltage variation at the output of operational amplifier 16 or output terminal 7 is e.sub.2 (t) and iii. the current flowing into the input of operational amplifier 16 is negligibly small, an equation (3) is established between the ionization chamber current variation i(t) and the voltage variations e.sub.1 (t) and e.sub.2 (t). ##EQU1## where EQU C.sub.i =C.sub.d +C.sub.c
r.sub.d is the dynamic resistance of logarithmic diode 14; the value of r.sub.d varies depending on the diode current I.sub.d which passes through the logarithmic diode.
The dynamic resistance r.sub.d is given by an equation (4) as is well known. ##EQU2## where V.sub.d is the diode voltage EQU I.sub.d =I+i(t).perspectiveto.I.
k is the Boltzmann's constant,
T is temperature in .degree.K. at the junction point of the logarithmic diode, and
q is elementary charge of electron.
Accordingly, the dynamic resistance r.sub.d varies from 26 .OMEGA. to 2.6.times.10.sup.10 .OMEGA. for the input current range of 10.sup.-3 A to 10.sup.-12 A.
If equation (3) is arranged with respect to e.sub.1 (S) and e.sub.2 (S) after the Laplace transformation, an equation (5) is obtained. ##EQU3##
The relationship between e.sub.1 (S) and e.sub.2 (S) can be expressed by means of the transfer function G.sub.o (S) of operational amplifier 16 which is given in equation (1), in an equation (6). EQU e.sub.2 (S)=G.sub.o (S)e.sub.1 (S) (6)
Therefore, ##EQU4##
The relationship between i(S) and e.sub.2 (S) is obtained from equation (5) after e.sub.1 (S) in equation (5) is substituted by e.sub.1 (S) in equation (7) and, thereafter, the transfer function G.sub.1 (S) as to the small current variation in the circuit shown in FIG. 2 is given in equation (8). ##EQU5##
When it is assumed that the small current variation i(t) varies stepwise, the following response conditions are considered to be the response of the output from the circuit shown in FIG. 2.
(1) over damping,
(2) critical damping,
(3) overshoot,
(4) damped oscillation or ringing,
(5) continuous oscillation, etc.
Conditions (1) and (2) are distinguished as stable responses and condition (3) to (5) as unstable responses.
Besides, when stepwise signals are applied to a circuit having a transfer function G(S) which is given by the form defined in an equation (9), as input to the circuit, EQU G(S)=(C/S.sup.2 +aS+b) (9)
if EQU a&gt;0 and b&gt;0 and (10) EQU a.sup.2 -4b.gtoreq.0, (11)
it is well known that the response in the output of the circuit becomes an over damping or a critical damping, in other words, the damping factor exceeds 1 or is 1.
When equation (8) is compared with equation (9) and the values corresponding to the coefficients a, b and c are sought, it is realized that a&gt;0 and b&gt;0. The values of a and b thus obtained are substituted in equation (11) and the value of the dynamic resistance r.sub.d given in equation (4) is substituted in equation (8) and at the same time it is assumed that (1+A/A).perspectiveto.1 (because A&gt;&gt;1) and I=Imax (the maximum value of the input current I). Under these conditions, a conditional equation (12) is obtained. ##EQU6## rdingly, equation (12) shows the relationship among the circuit constants which is necessary to obtain a stable step response having no overshoot nor ringing etc., in the logarithmic amplifying circuit of the prior art shown in FIG. 2. Under the condition that
Imax=1 mA
C.sub.i =3000 pF
A=10.sup.2, 10.sup.3 or 10.sup.5
the relationship that satisfies equation (3) between the minimum feedback capacitance C.sub.fmin and the unity gain frequency f.sub.T is shown in FIG. 3. It will be realized from the curves given in FIG. 3 that the minimum feedback capacitance needed C.sub.fmin is about 14700 pF with an operational amplifier having f.sub.T =1 MHz and C.sub.fmin is about 3300 pF with f.sub.T =10 MHz in order to obtain a stable response.
The time constant T.sub.O (63% value) of step response in the logarithmic amplifier of the prior art shown in FIG. 2 is nearly equal to a product of the dynamic resistance r.sub.d of the logarithmic diode and the capacitance C.sub.f of negative feedback capacitor 15 and, therefore, the following equation is obtained. EQU T.sub.O .perspectiveto.r.sub.d C.sub.f =2.6.times.10.sup.-2 C.sub.f /I (13)
The values of C.sub.fmin, already obtained, are substituted for the C.sub.f in equation (13) and the relationship between the response characteristics and the input current is obtained as shown in FIG. 4 (a) and (b) under the condition that the unity gain frequency f.sub.T of the operational amplifier is 1 MHz and 10 MHz, respectively. The relationship shows that T.sub.O is about 3.8 sec with f.sub.T =1 MHz and T.sub.O is about 0.86 sec even with f.sub.T =10 MHz at a small input current I such as 10.sup.-10 A. This means that an extremely slow response speed is merely obtained. Generally speaking since the limit of the unity gain frequency f.sub.T of an operational amplifier having high input resistance and with phase compensated, available under the present technique, is about 10 MHz, it can be said that the response characteristics given in FIG. 4 (b) is the boundary that can be obtained with a logarithmic measuring apparatus in the prior art.
As an explanation of the prior art, J. A. DE SHONG, JR. invented a method for improving the transient response over a wide range of input current by using a specially designed operational amplifier of which the forward gain phase shift never exceeds 45 degrees until the gain is reduced to unity or less and succeeded to reduce the response time of the logarithmic amplifier in a current range of low level. However, as can be seen from the explanation given in col. 6, lines 30 to 55 of the specification of U.S. Pat. No. 2,818,504 to J. A. DE SHONG, JR. and the frequency characteristics shown in FIG. 3 of DE SHONG, a response with completely no overshoot with respect to step inputs over the whole input current range cannot be obtained purely through the inventor's method. Therefore, in order to satisfy the condition of "no overshoot", it is necessary to reduce the closed loop phase shift by connecting a feedback capacitor in parallel diode 118 shown in FIG. 1 of DE SHONG, as similar to the prior art stated above and a defect is recognized with the invention in that ordinary phase-compensated-operational amplifiers of integrated circuits available in the market cannot be used because the forward gain phase shift up to 90 degrees is generally permissible in the ordinary operational amplifiers.